Characterizing the interpretation of set theory in Martin-L¨of type theory Sergei Tupailo∗
Michael Rathjen
Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, England {rathjen, s.tupailo}@maths.leeds.ac.uk
Abstract Constructive Zermelo-Fraenkel set theory, CZF, can be interpreted in Martin-L¨ of type theory via the so-called propositions-as-types interpretation. However, this interpretation validates more than what is provable in CZF. We now ask ourselves: is there a reasonably simple axiomatization (by a few axiom schemata say) of the set-theoretic formulae validated in Martin-L¨ of type theory? The answer is yes for a large collection of statements called the mathematical formulae. The validated mathematical formulae can be axiomatized by suitable forms of the axiom of choice. Keywords: Constructive Set Theory, Mart.
1
Introduction
The general topic of Constructive Set Theory (CST ) originated in John Myhill’s endeavour (see [17]) to discover a simple formalism that relates to Bishop’s constructive mathematics as classical Zermelo-Fraenkel Set Theory with the axiom of choice relates to classical Cantorian mathematics. CST provides a standard set theoretical framework for the development of constructive mathematics in the style of Errett Bishop [8]. One of the hallmarks of constructive set theory is that it possesses (due to Aczel [1, 2, 3]) a canonical interpretation in Martin-L¨of’s intuitionistic type theory (see [13, 14]) which is considered to be the most acceptable foundational framework of ideas that make precise the constructive approach to mathematics. The interpretation employs the Curry-Howard ‘propositions-as-types’ idea in that the axioms of constructive set theory get interpreted as provably inhabited types. The particular system of set theory for which Aczel gave a type-theoretic interpretation is actually a modification of Myhill’s system referred to as Constructive Zermelo-Fraenkel Set Theory, CZF. The interpretation of CZF in type theory (notated as ML1V) not only validates all the theorems of CZF but many other interesting set-theoretic statements as well. Ideally, one would like to have a characterization of these statements and determine an extension CZF∗ of CZF which deduces exactly the set-theoretic statements validated in the pertaining type theory ML1V. It will turn out that the search for CZF∗ amounts to finding the “strongest” version of the axiom of choice that is validated in ML1V. In addition to the axioms of CZF, Aczel also interpreted the Regular Extension Axiom, REA, which ensures the existence of many inductively defined sets. The particular type system that is sufficient for interpreting CZF + REA has been denoted by ML1WV. We shall also pursue the question of characterizing the set-theoretic statements validated in ML1WV. However, rather than giving a characterization of all set-theoretic statements validated in Martin-L¨of type theory, we shall restrict attention to a collection of formulae dubbed mathematical formulae which includes all the statements of workaday mathematics. The idea behind these formulae is that the sets of ordinary mathematics are of rank < ω + ω in the cumulative hierarchy. Roughly speaking, the mathematical formulae are bounded formulae with parameters in Vω+ω . We shall also consider the wider collection of generalized ∗ Research
supported by United Kingdom Engineering and Physical Sciences Research Council Grant GR/R 15856/01.
1
mathematical formulae which from the point of view of ZFC is concerned with sets of rank < ℵω . The main results of the paper are expressed in terms of the two choice principles ΠΣ−AC and ΠΣW−AC. Theorem 1.1 (Cf. 7.7) Let ψ be a mathematical sentence and let θ be a generalized mathematical sentence. Then the following hold: (i) CZF + ΠΣ−AC ` ψ if and only if ψ is validated in ML1V. (ii) CZF + REA + ΠΣW−AC ` θ if and only if θ is validated in ML1WV. The presentation of constructive mathematics in Martin-L¨of type theory is an obvious option for the constructive mathematician. However, it has the drawback that the syntactical apparatus is rather overpowering and that there is no extensive tradition of presenting mathematics in a type theoretic setting. This can be avoided by keeping to the set theoretical language. Constructive set theory is distinctive in that it uses the same language as classical set theory and it thus has the advantage that the ideas, conventions and practise of the set theoretical presentation of ordinary mathematics can be used also in constructive set theory. Theorem 1.1 sheds light on how these two approaches to constructive mathematics are related to each other. The plan for the paper is as follows: Section 2 discusses choice principles in constructive set theory. After briefly reviewing choice principles which have always featured prominently in constructive accounts of mathematics (axioms of countable choice and dependent choices) we explore the “strongest” versions of choice that can be validated in type theory, notably ΠΣ−AC and ΠΣW−AC. Sections 3 and 4 are concerned with interpreting constructive set theory in itself via a formulae-as-classes interpretation. This is done for bounded formulae in section 3 and for arbitrary formulae in section 4 via a notion of extended set recursive functions (building on [21]). Section 5 deals with the question of how the formulae-as-classes interpretation can be characterized via an inner model construction on the basis of ΠΣ−AC and ΠΣW−AC, respectively. Section 6 features interpretations of type theory in set theory also drawing on the notion of extended set recursive functions. In section 7 we are in a position to prove the main result Theorem 1.1. The last section presents some results about existential definability in theories with ΠΣ−AC and ΠΣW−AC. Notation. We will use hx, yi to notate the ordered pair of x and y. We use Fun(f ) to express that f is a function. dom(f ) and ran(f ) denote the domain and the range of f , respectively. f : A → B is used to convey that f is a function with dom(f ) = A and ran(f ) ⊆ B.
2
The axiom of choice in constructive set theories
Among the axioms of set theory, the axiom of choice is distinguished by the fact that is it the only one that one finds mentioned in workaday mathematics. In the mathematical world of the beginning of the 20th century, discussions about the status of the axiom of choice were important. In 1904 Zermelo proved that every set can be well-ordered by employing the axiom of choice. While Zermelo argued that it was selfevident, it was also criticized as an excessively non-constructive principle by some of the most distinguished analysts of the day, notably Borel, Baire, and Lebesgue. At first blush this reaction against the axiom of choice utilized in Cantor’s new theory of sets is surprising as the French analysts had used and continued to use choice principles routinely in their work. However, in the context of 19th century classical analysis only the Axiom of Dependent Choices, DC, is invoked and considered to be natural, while the full axiom of choice is unnecessary and even has some counterintuitive consequences. Unsurprisingly, the axiom of choice does not have a unambiguous status in constructive mathematics either. On the one hand it is said to be an immediate consequence of the constructive interpretation of the quantifiers. Any proof of ∀x ∈ A ∃y ∈ B φ(x, y) must yield a function f : A → B such that ∀x ∈ A φ(x, f (x)). This is certainly the case in Martin-L¨of’s intuitionistic theory of types. On the other hand, it has been observed that the full axiom of choice cannot be added to systems of extensional constructive set theory without yielding constructively unacceptable cases of excluded middle (see [9]). In extensional intuitionistic set theories, a proof of a statement ∀x ∈ A ∃y ∈ B φ(x, y), in general, provides only a function F , which when fed a proof p witnessing x ∈ A, yields F (p) ∈ B and φ(x, F (p)). Therefore, in the main, such an F cannot be rendered a function of x alone. Choice will then hold over sets which have a canonical proof function, where a 2
constructive function h is a canonical proof function for A if for each x ∈ A, h(x) is a constructive proof that x ∈ A. Such sets having natural canonical proof functions “built-in” have been called bases (cf. [24], p. 841). The particular form of constructivism adhered to in this paper is Martin-L¨of’s intuitionistic type theory (cf. [13, 14]). Set-theoretic choice principles will be considered as constructively justified if they can be shown to hold in the interpretation in type theory. Moreover, looking at set theory from a type-theoretic point of view has turned out to be valuable heuristic tool for finding new constructive choice principles. For more information on choice principles in the constructive context see [20].
2.1
Some constructive choice principles
In many a text on constructive mathematics, axioms of countable choice and dependent choices are accepted as constructive principles. This is, for instance, the case in Bishop’s constructive mathematics (cf. [8]) as well as Brouwer’s intuitionistic analysis (cf. [24], Ch. 4, Sect. 2). Myhill also incorporated these axioms in his constructive set theory [17]. The weakest constructive choice principle we shall consider is the Axiom of Countable Choice, ACω , i.e. whenever F is a function with domain ω such that ∀i ∈ ω ∃y ∈ F (i), then there exists a function f with domain ω such that ∀i ∈ ω f (i) ∈ F (i). A mathematically very useful axiom to have in set theory is the Dependent Choices Axiom, DC, i.e., for all formulae ψ, whenever (∀x ∈ a) (∃y ∈ a) ψ(x, y) and b0 ∈ a, then there exists a function f : ω → a such that f (0) = b0 and (∀n ∈ ω) ψ(f (n), f (n + 1)). Even more useful is the Relativized Dependent Choices Axiom, RDC. It asserts that for arbitrary formulae φ and ψ, whenever £ ¡ ¢¤ ∀x φ(x) → ∃y φ(y) ∧ ψ(x, y) and φ(b0 ), then there exists a function f with domain ω such that f (0) = b0 and £ ¤ (∀n ∈ ω) φ(f (n)) ∧ ψ(f (n), f (n + 1)) .
2.2
Operations on sets
The interpretation of constructive set theory in type theory not only validates all the theorems of CZF (resp. CZF + REA) but many other interesting set-theoretic statements, including several new choice principles which will be described next. To state these principles we need to introduce various operations on classes. Remark 2.1 Class notation: In doing mathematics in CZF we shall exploit the use of class notation and terminology, just as in classical set theory. Given a formula φ(x) there may not exist a set of the form {x : φ(x)}. But there is nothing wrong with thinking about such collection. So, if φ(x) is a formula in the language of set theory we may form a class {x : φ(x)}. We allow φ(x) to have free variables other than x, which are considered parameters upon which the class depends. Informally, we call any collection of the form {x : φ(x)} a class. However formally, classes do not exist, and expressions involving them must be thought of as abbreviations for expressions not involving them. Classes A, B are defined to be equal if ∀x[x ∈ A ↔ x ∈ B]. We may also consider an augmentation of the language of set theory whereby we allow atomic formulas of the form y ∈ A and A = B with A, B being classes. There is no harm in taking such liberties as any such formula can be translated back into the official language of set theory by re-writing y ∈ {x : φ(x)} and {x : φ(x)} = {y : ψ(y)} as φ(y) and ∀z [φ(z) ↔ ψ(z)], respectively (with z not in φ(x) and ψ(y)). Definition 2.2 Let CZFExp denote the modification of CZF with Eponentiation in place of Subset Collection.
3
Remark 2.3 In all the results of this paper, CZF could be replaced by CZFExp , that is to say, for the purposes of this paper it is enough to assume Exponentiation rather than Subset Collection. However, in what follows we shall not point this out again. Definition 2.4 (CZF) Q If A is a set and Bx are classes for all x ∈ A, we define a class x∈A Bx by: Y [ Bx := {f : A → Bx | ∀x∈A(f (x) ∈ Bx )}. x∈A
(1)
x∈A
P If A is a class and Bx are classes for all x ∈ A, we define a class x∈A Bx by: X Bx := {hx, yi | x ∈ A ∧ y ∈ Bx }.
(2)
x∈A
If A is a class and a, b are sets, we define a class I(A, a, b) by: I(A, a, b) := {z ∈ 1 | a = b ∧ a, b ∈ A}.
(3)
If A is a class and for each a ∈ A, Ba is a set, then Wa∈A Ba is the smallest class Y such that whenever a ∈ A and f : Ba → Y , then ha, f i ∈ Y . Lemma 2.5 (CZF) P Q If A,B,a,b are sets and Bx are sets for all x ∈ A, then x∈A Bx , x∈A Bx and I(A, a, b) are sets. S Proof. First of all, we need to prove that x∈A Bx is a set. Indeed, g = {{x, {x, Bx }} | x ∈ A}, and so SS g = {z, x, Bx | z ∈ x, x ∈ A} is a set by Union. Now [[ [[ ran(g) = {y ∈ g | ∃x∈ g (hx, yi ∈ g)} S S Separation and Union. and x∈A Bx = ran(g) are sets by Bounded S 1: The class of all functions from A to x∈A Bx is a set by Exponentiation and [ Y Bx := {f : A → Bx | ∀x∈A(f (x) ∈ Bx )} x∈A
x∈A
is a set by Bounded Separation, since ∀x∈A(f (x) ∈ Bx ) can be rewritten as ∀x∈A ∃y ∈ran(f )∃y 0 ∈ran(g)(hx, yi ∈ f ∧ hx, y 0 i ∈ g ∧ y ∈ y 0 ). S 2: Using from above that x∈A Bx is a set, by Pairing, Union and Replacement we obtain a set [ [ A× Bx = {hx, yi | x ∈ A ∧ y ∈ Bx }. x∈A
Now, the set
X
Bx := {hx, yi∈A ×
x∈A
x∈A
[
Bx | x ∈ A ∧ y ∈ Bx }
x∈A
exists by Bounded Separation, since x ∈ A ∧ y ∈ Bx can be rewritten as x ∈ A ∧ ∃y 0 ∈ran(g)(hx, y 0 i ∈ g ∧ y ∈ y 0 ). 3: I(A, a, b) is a set by Bounded Separation.
2
Lemma 2.6 (CZF + REA) If A is a set and Bx is a set for all x ∈ A, then Wa∈A Ba is a set. Proof. This follows from [3], Corollary 5.3.
2 4
2.3
Inductively defined classes
In the following we shall introduce several inductively defined classes, and, moreover, we have to ensure that such classes can be formalized in CZF. We define an inductive definition to be a class of ordered pairs. If Φ is an inductive definition and hx, ai ∈ Φ then we write x Φ a and call xa an (inference) step of Φ, with set x of premisses and conclusion a. For any class Y , let ΓΦ (Y )
=
©
¡ a | ∃x x ⊆ Y
∧
x a
Φ
¢ª .
The class Y is Φ-closed if ΓΦ (Y ) ⊆ Y . Note that Γ is monotone; i.e. for classes Y1 , Y2 , whenever Y1 ⊆ Y2 , then Γ(Y1 ) ⊆ Γ(Y2 ). We define the class inductively defined by Φ to be the smallest Φ-closed class. The main result about inductively defined classes states that this class, denoted I(Φ), always exists. Lemma 2.7 (CZF) (Class Inductive Definition Theorem) For any inductive definition Φ there is a smallest Φ-closed class I(Φ). Moreover, call a set G of ordered pairs good if ha, yi ∈ G ⇒ y ∈ ΓΦ (G∈a ).
(∗) where Letting J =
S
G∈a = {y 0 | ∃x∈a hx, y 0 i ∈ G}. {G | G is good} and J a = {x | ha, xi ∈ J}, it holds [ I(Φ) = J a, a
and for each a, J a = ΓΦ (
[
J x ).
x∈a
J is uniquely determined by the above, and its stages J a will be denoted by ΓaΦ . Proof. [2], section 4.2 or [4], Theorem 5.1.
2
Lemma 2.8 (CZF) There exists a smallest ΠΣ-closed class, i.e. a smallest class Y such that the following holds: (i) n ∈ Y for all n ∈ N; (ii) ωQ∈ Y; P (iii) x∈A Bx ∈ Y and x∈A Bx ∈ Y whenever A ∈ Y and Bx ∈ Y for all x ∈ A. Likewise, there exists a smallest ΠΣI-closed class, i.e. a smallest class Y∗ , which, in addition to the closure conditions (i)–(iii) above, satisfies: (iv) I(A, a, b) ∈ Y∗ whenever A ∈ Y∗ and a, b ∈ A. Proof. The classes Y and Y∗ are inductively defined, and therefore exist by Lemma 2.7. To be precise, the respective inductive definitions of these classes are given by the classes Φ1 , . . . , Φ5 consisting of the following pairs: (i) (ii) (iii)
n
Φ1
, for all n ∈ N;
ω
Φ2
;
{dom(g)} ∪ ran(g) Q x∈A g(x)
Φ3
, for all functions g with dom(g) = A; 5
(iv)
{dom(g)} ∪ ran(g) P x∈A g(x)
(v)
{A} I(A, a, b)
Φ5
Φ4
, for all functions g with dom(g) = A;
, if a, b ∈ A.
(Clause (v) is only needed to define Y∗ .)
2
Lemma 2.9 (CZF + REA) There exists a least ΠΣW-closed class, i.e. a smallest class Yw that in addition to the clauses (i),(ii),(iii) of Lemma 2.8 satisfies: (vi) Wa∈A Ba ∈ Yw whenever A ∈ Yw and Bx ∈ Yw for all x ∈ A. ∗ Likewise, there exists a smallest ΠΣWI-closed class, i.e. a least class Yw , which, in addition to the closure conditions above, satisfies clause (iv) of Lemma 2.8.
Proof. Virtually the same as for Lemma 2.8.
2.4
2
Strong choice principles
Definition 2.10 The ΠΣ-generated sets are the sets in the smallest ΠΣ-closed class, i.e. Y. Similarly one defines the ΠΣI, ΠΣW and ΠΣWI-generated sets. A set P is a base if for any P -indexed family (Xa )a∈P of inhabited sets Xa , there exists a function f with domain P such that, for all a ∈ P , f (a) ∈ Xa . ΠΣ−AC is the statement that every ΠΣ-generated set is a base. Similarly one defines the axioms ΠΣI−AC, ΠΣWI−AC, and ΠΣW−AC. Lemma 2.11
(i) (CZF) For every A ∈ Y∗ there exists a B ∈ Y with a bijection h : B → A.
∗ there exists a B ∈ Yw with a bijection h : B → A. (ii) (CZF + REA) For every A ∈ Yw
Proof. See the lemma following Theorem 3.7 in [3]. Corollary 2.12
2
(i) (CZF) ΠΣ−AC and ΠΣI−AC are equivalent.
(ii) (CZF + REA) ΠΣW−AC and ΠΣWI−AC are equivalent. Proof. ΠΣI−AC obviously implies ΠΣ−AC, since Y ⊆ Y∗ . To prove the converse, assume ΠΣ−AC, A ∈ Y∗ , and ∀x ∈ A∃yϕ(x, y), where ϕ is a formula of CZF. Take a B and a bijection h : A → B which exists by the previous Lemma; then ∀x∈B∃y ϕ(h−1 (x), y). By ΠΣ−AC, ∃f : B → V ∀x∈B ϕ(h−1 (x), f (x)). This yields
¡ ¢ ∀x∈A ϕ h−1 ◦ h(x), f ◦ h(x)
¡ ¢ so that ∀x∈A ϕ x, f ◦ h(x) . The proof of (ii) is similar.
3
2
Interpreting bounded formulae as sets
Notation. For sets x and y, we define sup(x, y) as hx, yi. If α = sup(A, f ), where f is a function with domain A, we define α ¯ := A and α ˜ := f . Definition 3.1 (CZF) By Lemma 2.7 we define classes V(Y∗ ) and H(Y∗ ) by the following rules: a ∈ Y∗ f : a → V(Y∗ ) , sup(a, f ) ∈ V(Y∗ ) 6
(4)
a ∈ Y∗ f : a → H(Y∗ ) . ran(f ) ∈ H(Y∗ )
(5)
The classes V(Y) and H(Y) are defined in the same vein by replacing Y∗ by Y in the foregoing clauses. Definition 3.2 (CZF) . The (class) functions = : V(Y∗ ) × V(Y∗ ) → Y∗ and ∈˙ : V(Y∗ ) × V(Y∗ ) → Y∗ are defined by recursion YX . YX . as follows: . ˜ ˜ = (α, β) is = (α ˜ (x), β(y)) × = (˜ α(x), β(y)), (6) x∈α ¯ y∈β¯
¯ y∈β¯ x∈α
X . ˜ ∈˙ (α, β) is = (α, β(y)).
(7)
y∈β¯
Definition 3.3 (CZF + REA) ∗ ∗ In the same vein as in Definitions 3.1 and 3.2 we define classes V(Yw ), V(Yw ), H(Yw ), H(Yw ), and (class) . ∗ ∗ ∗ ∗ ∗ ∗ ∗ ˙ . functions = : V(Yw ) × V(Yw ) → Yw and ∈ : V(Yw ) × V(Yw ) → Yw by replacing Y∗ with Yw . . Convention. We will write α = β and α ∈˙ β for = (α, β) and ∈˙ (α, β), respectively. Lemma 3.4
(i) (CZF) H(Y) = H(Y∗ ).
∗ (ii) (CZF + REA) H(Yw ) = H(Yw ).
Proof. (i): Plainly, we have H(Y) ⊆ H(Y∗ ). To show H(Y∗ ) ⊆ H(Y), we shall draw on Lemma 2.7. Let ΓH(Y∗ ) be the operator that inductively defines H(Y∗ ) so that H(Y∗ ) =
[
ΓaH(Y∗ ) .
a
S Proceeding by set induction on a, we show that ΓaH(Y∗ ) ⊆ H(Y). So assume that b∈a ΓbH(Y∗ ) ⊆ H(Y) and S suppose g : A → b∈a ΓbH(Y∗ ) , where A ∈ Y∗ . Owing to Lemma 2.11 there exists B ∈ Y and a bijection h : B → A. Then we have g ◦ h : Y → H(Y), and thus ran(g) = ran(g ◦ h) ∈ H(Y). Consequently, ΓaH(Y∗ ) ⊆ H(Y). (ii) is proved similarly. 2 Definition 3.5
(i) (CZF) The mapping ` : V(Y∗ ) → H(Y∗ ) is defined by recursion via `(sup(a, f )) :=
{`(f (i)) | i ∈ a} = ran(` ◦ f ).
(8)
∗ ∗ (ii) (CZF + REA) The mapping `w : V(Yw ) → H(Yw ) is defined by recursion via
`w (sup(a, f )) := Lemma 3.6
(i) (CZF + ΠΣ−AC)
(ii) (CZF + ΠΣW−AC)
{`w (f (i)) | i ∈ a} = ran(`w ◦ f ).
(9)
` is surjective.
`w is surjective.
Proof. By induction on the inductive generation of H(Y∗ ), we prove ∀x∈H(Y∗ )∃z ∈V(Y∗ )(`(z) = x). Let x ∈ H(Y∗ ). Then x = ran(g) for some function g : A → H(Y∗ ), where A ∈ Y∗ . Inductively we have ∀u ∈ A∃c ∈ V(Y∗ )(`(c) = g(u)). By ΠΣI−AC, which by Corollary 2.12 is equivalent to ΠΣ−AC, there is a function f : A → V(Y∗ ) such that ∀u ∈ A(`(f (u)) = g(u)). Note that sup(A, f ) ∈ V(Y∗ ). Hence `(sup(A, f )) = ran(` ◦ f ) = ran(g) = x. (ii) is proved similarly. 2 7
Lemma 3.7 (CZF + ΠΣ−AC) Let α, β ∈ V(Y∗ ). Then we have . (i) ∃i∈(α = β) ⇔ `(α) = `(β). (ii) ∃i∈(α ∈˙ β) ⇔ `(α) ∈ `(β). Proof. (i) is proved by induction on α and β: . ∃i∈(α = β)
. ∃uv(hu, vi ∈ (α = β)) YX YX . ˜ . ˜ ∃u∈ (˜ α(x) = β(y)) ∧ ∃v ∈ (˜ α(x) = β(y))
⇐⇒ ⇐⇒
x∈α ¯ y∈β¯
¯ y∈β¯ x∈α
. ˜ . ˜ ¯ ¯ ∀x∈ α ¯ ∃y ∈ β∃i∈(˜ α(x) = β(y)) ∧ ∀y ∈ β∃x∈ α ¯ ∃j ∈(˜ α(x) = β(y))
ΠΣI−AC
⇐⇒ IH
¯ α(x)) = `(β(y))) ˜ ¯ ˜ ∀x∈ α ¯ ∃y ∈ β(`(˜ ∧ ∀y ∈ β∃x∈ α ¯ (`(˜ α(x)) = `(β(y))) ˜ ran(` ◦ α) ˜ = ran(` ◦ β) `(α) = `(β).
⇐⇒ ⇐⇒ ⇐⇒ (ii) now follows from (i):
∃i∈(α ∈˙ β)
. ˜ ¯ ∈(α = ⇐⇒ ∃y ∈ β∃j β(y)) (i) ¯ ˜ ⇐⇒ ∃y ∈ β(`(α) = `(β(y))) ˜ ⇐⇒ `(α) ∈ ran(` ◦ β) ⇐⇒ `(α) ∈ `(β).
2 Lemma 3.8 (CZF + ΠΣW−AC) ∗ Let α, β ∈ V(Yw ). Then we have . (i) ∃i∈(α = β) ⇔ `w (α) = `w (β). (ii) ∃i∈(α ∈˙ β) ⇔ `w (α) ∈ `w (β). Proof. The same as for Lemma 3.7.
2
Definition 3.9 (CZF) For any set A and class B we define: A→B
as
Q x∈A
B.
(10)
For any classes A and B we define: A×B A+B
P as Px∈A B, as x∈2 Cx ,
where C0 = A and C1 = B.
(11)
Definition 3.10 A V(Y∗ )-assignment is a mapping M : Var → V(Y∗ ), where Var is the set of variables of the language. M(a) will also be denoted by aM . If M is a V(Y∗ )-assignment, u is a variable, and d ∈ V(Y∗ ), we define a V(Y∗ )-assignment M(u|d) by ½ M(v) if v is a variable other than u M(u|d)(v) = d if v is u. Sometimes, when an assignment M is fixed, we will omit the subscript
8
M.
Definition 3.11 (CZF) To any bounded formula θ ∈ L∈ and V(Y∗ )-assignment M we shall assign a set k θ k M . Since we have already used the symbol “→” for function spaces we shall denote the conditional by “ ⊃”. The recursive definition of k θ k M is given in the table below: θ ∈ L∈
k θ kM
⊥
0
a=b
. aM = bM
a∈b
aM ∈˙ bM
θ0 ∧ θ1
k θ0 k M × k θ1 k M
θ0 ∨ θ1
k θ0 k M + k θ1 k M
θ0 ⊃ θ1
k θ0 k M → k θ1 k M Q k ψ k M(v|g x∈a aM (x)) P M k ψ k x∈αM M(v|g aM (x))
∀v ∈a ψ ∃v ∈a ψ
Lemma 3.12 (CZF) For every bounded θ ∈ L∈ and V(Y∗ )-assignment M, k θ k M ∈ Y∗ . Proof. This is proved by induction on θ using Lemma 2.8 and Definitions 3.2 and 3.9.
2
Lemma 3.13 (CZF + REA) ∗ A V(Yw )-assignment is defined similarly as a V(Y∗ )-assignment in Definition 3.10. Likewise, as in ∗ )-assignment M we assign a set k θ k M . We Definition 3.11, to any bounded formula θ ∈ L∈ and V(Yw ∗ ∗ then have, for every bounded θ ∈ L∈ and V(Yw )-assignment M, k θ k M ∈ Yw . Proof. This is proved as Lemma 3.12.
2
Theorem 3.14 (CZF + ΠΣ−AC) For every bounded θ ∈ L∈ and V(Y∗ )-assignment M, ∃i∈k θ k M
⇔
θ`(M) ,
where θ`(M) denotes the result of replacing every free variable a of θ by `(aM ). Proof is by induction on θ. If θ is ⊥, the assertion is obvious. If θ is a = b or a ∈ b, the assertion follows from Lemma 3.7. Assume θ is θ0 ∧ θ1 . Then: IH
`(M)
∃i∈k θ0 ∧ θ1 k M ⇐⇒ ∃u∈k θ0 k M ∧ ∃v ∈k θ1 k M ⇐⇒ θ0 Assume θ is θ0 ∨ θ1 . Then: ∃i∈k θ0 ∨ θ1 k M
`(M)
∧ θ1
.
£ ¤ ⇐⇒ ∃u∈2 ∃v (u = 0 ∧ v ∈ k θ0 k M ) ∨ (u = 1 ∧ v ∈ k θ1 k M ) ⇐⇒ ∃u [u = 0 ∧ ∃v ∈k θ0 k M ] ∨ ∃u [u = 1 ∧ ∃v ∈k θ1 k M ] ⇐⇒ ∃v ∈k θ0 k M ∨ ∃v ∈k θ1 k M IH
`(M)
⇐⇒ θ0
`(M)
∨ θ1
.
Assume θ is (θ0 ⊃ θ1 ). Then: ∃f ∈k θ0 → θ1 k M
⇐⇒ ⇐⇒ ΠΣI−AC
⇐⇒ ⇐⇒ IH
⇐⇒
∃f ∈(k θ0 k M 7→ k θ1 k M ) ∃f (Fun[f ] ∧ dom(f ) = k θ0 k M ∧ ∀y ∈k θ0 k M (f (y) ∈ k θ1 k M )) ∀y ∈k θ0 k M ∃i∈k θ1 k M ∃y ∈k θ0 k M ⊃ ∃i∈k θ1 k M `(M)
θ0
`(M)
⊃ θ1
. 9
Assume θ is ∀v ∈a ψ. Then: ∃f ∈k ∀v ∈a ψ k M
⇐⇒
∃f ∈
Y
k ψ k M(v|g aM (x))
x∈aM
⇐⇒ ΠΣI−AC
⇐⇒ IH
⇐⇒ ⇐⇒
¡ ¢ ∃f Fun(f ) ∧ dom(f ) = aM ∧ ∀x∈aM (f (x) ∈ k ψ k M(v|g aM (x)) ) ∀x∈aM ∃i∈k ψ k M(v|g aM (x)) aM (x)) ∀x∈aM ψ `(M(v|g (∀v ∈a ψ)`(M) .
Assume θ is ∃v ∈a ψ. Then: ∃d∈k ∃v ∈a ψ k M
⇐⇒ ∃d∈
X
k ψ k M(v|g aM (x))
x∈aM
⇐⇒ ∃x∈aM ∃s∈k ψ k M(v|g aM (x)) IH
aM (x)) ⇐⇒ ∃x∈aM ψ `(M(v|g `(M) ⇐⇒ (∃v ∈a ψ) .
2 Theorem 3.15 (CZF + REA + ΠΣW−AC) ∗ For every bounded θ ∈ L∈ and V(Yw )-assignment M, ∃i∈k θ k M
⇔
θ`w (M) ,
where θ`w (M) denotes the result of replacing every free variable a of θ by `w (aM ). Proof is by induction on θ as in the previous Theorem 3.14.
4
2
The formulae-as-classes interpretation for arbitrary formulae
In order to reflect within CZF the formulae-as-classes interpretation for arbitrary set-theoretic formulae and judgements of ML1 V, we would need to represent large types ΠΣ-generated on top of V(Y∗ ). The language of CZF is not rich enough to do it in a straightforward way. To remedy this we utilize a special notion of set recursive partial function developed in [21].
4.1
Extended E-recursive functions
We would like to have unlimited application of sets to sets, i.e. we would like to assign a meaning to the symbol {a}(x) where a and x are sets. In generalized recursion theory this is known as E-recursion or set recursion (see, e.g., [18] or [23, Ch.X]). However, we shall introduce an extended notion of E-computability, christened E℘ -computability, rendering the function exp(a, b) = a b is computable as well, (where a b denotes the set of all functions from a to b). Moreover, the constant function with value ω is taken as an initial function in E℘ -computability. From a classical standpoint, E℘ -computability is related to power recursion, where the power set operation is regarded to be an initial function. The latter notion has been studied by Moschovakis [15] and Moss [16]. There is a lot of leeway in setting up E℘ -recursion. The particular schemes we use are especially germane to our situation. Our construction will provide a specific set-theoretic model for the elementary theory of operations and numbers EON (see, e.g., [7, VI.2], or the theory APP as described in [24, Ch.9, Sect.3]). We utilize encoding of finite sequences of sets by the pairing function h , i. Definition 4.1 (CZF) ¯ ω First, we select distinct non-zero natural numbers k, s, p, p0 , p1 , sN , pN , dN , 0, ¯ , π, σ, pl, i, fa, and 10
ab which will provide indices for special E℘ -recursive partial (class) functions. Inductively we shall define a class E of triples he, x, yi. Rather than “he, x, yi ∈ E”, we shall write “{e}(x) ¥ y”, and moreover, if n > 0, we shall use {e}(x1 , . . . , xn ) ¥ y to convey that {e}(x1 ) ¥ he, x1 i ∧ {he, x1 i}(x2 ) ¥ he, x1 , x2 i ∧ . . . ∧ {he, x1 , . . . , xn−1 i}(xn ) ¥ y. We shall say that {e}(x) is defined, written {e}(x) ↓, if {e}(x) ¥ y for some y. Let N := ω. E is defined by the following clauses (inference steps): {k}(x, y) {s}(x, y, z) {p}(x, y) {p0 }(x) {p1 }(x) {sN }(n) {pN }(0) {pN }(n + 1) {dN }(n, m, x, y) {dN }(n, m, x, y) ¯ {0}(x) {¯ ω }(x) {π}(x, g) {σ}(x, g) {pl}(x, y) {i}(x, y, z) {fa}(g, x) {ab}(e, a)
¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥
x {{x}(z)}({y}(z)) hx, yi (x)0 (x)1 n + 1 if n ∈ N 0 n if n ∈ N x if n, m ∈ N and n = m y if n, m ∈ N and n 6= m 0 ω Q Pz∈x g(z) if g is a (set-)function with dom(g) = x z∈x g(z) if g is a (set-)function with dom(g) = x x+y I(x, y, z) g(x) if g is a (set-)function and x ∈ dom(g) h if h is a (set-)function with dom(h) = a and ∀x∈a {e}(x) ¥ h(x).
Note that for {s}(x, y, z) to be defined it is required that {x}(z), {y}(z) and {{x}(z)}({y}(z)) be defined. The clause for s is thus to be read as a conjunction of the following clauses: {s}(x) ¥ hs, xi, {hs, xi}(y) ¥ hs, x, yi and, if there exist a, b, c such that {x}(z) ¥ a, {y}(z) ¥ b, {a}(b) ¥ c, then {hs, x, yi}(z) ¥ c. The constants fa and ab stand for function application and function abstraction, respectively. Lemma 4.2 (CZF) E is an inductively defined class and E is functional in that for all e, x, y, y 0 , he, x, yi ∈ E ∧ he, x, y 0 i ∈ E ⇒
y = y0 .
Proof. The inductive definition of E falls under the heading of Lemma 2.7. If {e}(x) ¥ y the uniqueness of y follows by induction on the stages (see Lemma 2.7) of that inductive definition. 2 Definition 4.3 Application terms are defined inductively as follows: ¯ ω ,π,σ,pl,i,fa,ab singled out in Definition 4.1 are application (i) The constants k,s,p,p0 ,p1 ,sN ,pN ,dN ,0,¯ terms; (ii) variables are application terms; (iii) if s and t are application terms then (st) is an application term. Definition 4.4 Application terms are easily formalized in CZF. However, rather than translating application terms into the set-theoretic language of CZF, we define the translation of expressions of the form t ' u, where t is an application term and u is a variable. The translation proceeds along the way that t was built up: [c ' u]∧ is c = u if c is a constant or a variable; [(st) ' u]∧ is ∃x∃y([s ' x]∧ ∧ [t ' y]∧ ∧ hx, y, ui ∈ E). 11
Abbreviations. For application terms s, t, t1 , . . . , tn we will use: s(t1 , . . . , tn ) st1 . . . tn t↓ (s ' t)∧ {x}(y) = z
as as as as as
a a a a a
shortcut shortcut shortcut shortcut shortcut
for for for for for
((. . . (st1 ) . . .)tn ) (parentheses associated to the left); s(t1 , . . . , tn ); ∃x(t ' x)∧ ; (t is defined ) s↓ ∨ t↓ ⊃ ∃x((s ' x)∧ ∧ (t ' x)∧ ); hx, y, zi ∈ E.
A closed application term is an application term that does not contain variables. If t is a closed application term and a1 , . . . , an , b are sets we use the abbreviation ¡ t(a1 , . . . , an ) ' b for ∃x1 . . . xn ∃y x1 = a1 ∧ . . . ∧ xn = an ∧ y = b ¢ ∧ [t(x1 , . . . , xn ) ' y]∧ . Definition 4.5 Every closed application term gives rise to a partial class function. A partial n-place (class) function Υ is said to be an E℘ -recursive partial function if there exists a closed application term tΥ such that dom(Υ)
=
{(a1 , . . . , an ) | tΥ (a1 , . . . , an ) ↓}
and for all for all sets (a1 , . . . , an ) ∈ dom(Υ), tΥ (a1 , . . . , an ) '
Υ(a1 , . . . , an ).
In the latter case, tΥ is said to be an index for Υ. If Υ1 , Υ2 are E℘ -recursive partial functions, then Υ1 (~a) ' Υ2 (~a) iff neither Υ1 (~a) nor Υ2 (~a) are defined, or Υ1 (~a) and Υ2 (~a) are defined and equal. The next two results can be proved in the theory APP and thus hold true in any applicative structure. Thence applicative structure above satisfies the Abstraction Lemma and Recursion Theorem (see e.g. [10] or [7]). Lemma 4.6 (Abstraction lemma, cf. [7, VI.2.2]) For every application term t[x] there exists an application term λx.t[x] with FV(λx.t[x]) := {x1 , . . . , xn } ⊆ FV(t[x])\{x} such that the following holds: ∀x1 . . . ∀xn (λx.t[x]↓ ∧ ∀y (λx.t[x])y ' t[y]). ¡ ¢ Proof. (i) λx.x is skk; (ii) λx.t is kt for t a constant or a variable other than x; (iii) λx.uv is s(λx.u) (λx.v). u t Lemma 4.7 (Recursion Theorem, cf. [7, VI.2.7]) There exists a closed application term rec such that for any f , x, recf ↓ ∧ recf x ' f (recf )x. Proof. Take rec to be λf.tt, where t is λyλx.f (yy)x.
u t
Corollary 4.8 For any E℘ -recursive partial function Υ there exists a closed application term τf ix such that τf ix ↓ and for all ~a, Υ(¯ e, ~a)
' τf ix (~a),
where τf ix ' e¯. Moreover, τf ix can be effectively (e.g. primitive recursively) constructed from an index for Υ.
12
4.2
Arbitrary formulae
Definition 4.9 (CZF) If B is a class and a, x are sets, we write {a}(x) ∈ B with the following meaning: {a}(x) ∈ B :⇔ ∃y({a}(x) = y ∧ y ∈ B).
(12)
Q If A is a class and Bx are classes for all x ∈ A, then we define a class e x∈A Bx in the following way: Y f Bx := {a | ∀x∈A({a}(x) ∈ Bx )}.
(13)
x∈A
For any classes A and B we define a class A f → B by Af → B := {a | ∀x∈A({a}(x) ∈ B)} =
Y f x∈A
B.
(14)
Definition 4.10 (CZF) . For every formula θ ∈ L∈ and V(Y∗ )-assignment M, we define a class k θ k M . The definition is given by the table below: .
θ ∈ L∈
k θ kM
⊥
0
a=b
. aM = bM
a∈b
aM ∈˙ bM
θ0 ∧ θ1
k θ0 k M × k θ1 k M
θ0 ∨ θ1
k θ0 k M + k θ1 k M
θ0 ⊃ θ1
k θ0 k M → k θ1 k M
.
.
.
.
.
if θ0 . k → k θ1 k M if θ0 f Q . x∈aM k ψ k M(v|g aM (x)) P . k ψ k x∈aM M(v|g aM (x)) Q . e k ψ k x∈V(Y ∗ ) M(v|x) P . k ψ k x∈V(Y ∗ ) M(v|x) . θ0 k M
θ0 ⊃ θ1 ∀v ∈a ψ ∃v ∈a ψ ∀vψ ∃vψ
is bounded is not bounded
Lemma 4.11 (CZF) . For every bounded formula θ and a V(Y∗ )-assignment M, k θ k M = k θ k M . Proof. This follows by induction on θ by comparing Definitions 3.11 and 4.10.
2
Definition 4.12 If θ(u1 , . . . , ur ) is a formula of L∈ all of whose free variables are among u1 , . . . , ur , and . α1 , . . . , αr ∈ V(Y∗ ), we shall use the shorthand k θ(α1 , . . . , αr ) k rather than k θ k M whenever M is an assignment satisfying M(ui ) = αi for 1 ≤ i ≤ r. In the special case when θ is a sentence we will simply write k θ k. We shall also use the following abbreviations: e ° θ(α1 , . . . , αr ) V(Y ) |= θ(α1 , . . . , αr )
iff iff
e ∈ k θ(α1 , . . . , αr ) k e ° θ(α1 , . . . , αr ) for some e
|=∗ θ(α1 , . . . , αr )
iff
V(Y∗ ) |= θ(α1 , . . . , αr ).
∗
For a set-theoretic formula θ(~u) we say that θ(α1 , . . . , αr ) is validated in V(Y∗ ) if we have produced a closed application term t such that t(~ α) ° θ(~ α) holds for all α ~ ∈ V(Y∗ ).
13
4.3
The formulae-as-classes interpretation for CZF
The rationale for the employment of the particular notion of extended E-recursive is revealed only in the proof of the following theorem. Theorem 4.13 ([21], Theorem 4.13) Let θ(u1 , . . . , ur ) be a formula of L∈ all of whose free variables are among u1 , . . . , ur . If CZF + ΠΣ−AC ` θ(u1 , . . . , ur ), then one can effectively construct an index of a E℘ -recursive partial function g such that CZFExp ` ∀α1 , . . . , αr ∈ V(Y∗ ) g(α1 , . . . , αr ) ∈ k θ(α1 , . . . , αr ) k. Recall that CZFExp denotes the modification of CZF with Exponentiation in place of Subset Collection. Proof. See [21], Theorem 4.13. The proof of 4.13 is rather long and requires close attention to the definition of indices of E℘ -recursive functions. u t
4.4
The formulae-as-classes interpretation for CZF + REA
As the reader may expect, the formulae-as-classes interpretation given for CZF above can be extended to CZF + REA also. The first step is to add the following condition to the definition of E℘ -recursive functions, giving rise to the E℘w -recursive functions: {w}(x, ¯ g) =
Wz∈x g(z) if g is a (set-)function with dom(g) = x,
where w ¯ is a “fresh” natural number. . ∗ )-assignment M, a class k θ k M as in Definition 4.10, One then defines for every formula θ ∈ L∈ and V(Yw where, however, the definition of the product Y f Bx := {a | ∀x∈A({a}(x) ∈ Bx )} (15) x∈A
is to be understood in the sense of
E℘w -recursive
functions. Correspondingly, we obtain the following result.
Theorem 4.14 ([21], Theorem 4.33) Let θ(u1 , . . . , ur ) be a formula of L∈ all of whose free variables are among u1 , . . . , ur . If CZF + REA + ΠΣW−AC ` θ(u1 , . . . , ur ), then one can effectively construct an index of a E℘w -recursive partial function g such that CZFExp + REA ` ∀α1 , . . . , αr ∈ V(Y∗ ) g(α1 , . . . , αr ) ∈ k θ(α1 , . . . , αr ) k, where CZFExp denotes the modification of CZF with Exponentiation in place of Subset Collection. Proof. See [21], Theorem 4.33. The proof builds on the proof of Theorem 4.13.
5
u t
The formulae-as-classes interpretation and validity in H(Y∗ )
The following considerations are reminiscent of Definition 3.8 and Theorem 3 of [25]. Definition 5.1 A formula is said to be CC if no unbounded quantifier in it occurs in the antecedent of an implication. Note that bounded as well as prenex (i.e. bounded preceded by a string of quantifiers) formulas are CC.
14
Theorem 5.2 (CZF + ΠΣ−AC) For every θ ∈ L∈ and any V(Y∗ )-assignment M, if θ is CC, then .
∃i∈k θ k M
⇒
θ`(M) ,
where θ`(M) denotes the result of replacing each free variable a of θ by `(aM ) and each unbounded quantifier Qx of θ by Qx∈H(Y∗ ). The proof is by induction on θ. If θ is an atom, the assertion follows from Lemma 4.11 and Theorem 3.14. If θ is a conjunction or disjunction, then the assertion follows easily from the IH. .
`(M)
Suppose θ is θ0 → θ1 and x ∈ k θ k M . Since θ ∈ CC, θ0 must be bounded. If θ0 , then by Theorem 3.14 . `(M) `(M) . As a result, θ . ∃i ∈ k θ0 k M , and thus x(i) ∈ k θ1 k M , which by the IH yields θ1 Assume θ is ∀v ∈a ψ. Then we have: .
∃f ∈k ∀v ∈a ψ k M
⇐⇒ ∃f ∈
Y
.
k ψ k M(v|g aM (x))
x∈aM
¡ £ ¢¤ . ⇐⇒ ∃f Fun[f ] ∧ dom(f ) = aM ∧ ∀x∈aM f (x) ∈ k ψ k M(v|g aM (x)) =⇒
.
∀x∈aM ∃y ∈k ψ k M(v|g aM (x))
IH
aM (x)) =⇒ ∀x∈aM ψ `(M(v|g ⇐⇒ (∀v ∈a ψ)`(M) .
Assume θ is ∃v ∈a ψ. Then: .
∃d ∈ k ∃v ∈a ψ k M
⇐⇒ ∃d∈
X
.
k ψ k M(v|j)
j∈aM .
⇐⇒ ∃j ∈aM ∃s∈k ψ k M(v|j) IH
aM (j)) =⇒ ∃j ∈aM ψ `(M(v|g ⇐⇒ (∃v ∈a ψ)`(M) .
Assume θ is ∀v ψ. Then we have: .
∃a ∈ k ∀v ψ k M
⇐⇒ ∃a ∈
Y f
.
x∈V(Y ∗ ) ¡ ∗
k ψ k M(v|x) .
⇐⇒ ∃a ∀x∈V(Y ) {a}(x) ∈ k ψ k M(v|x) =⇒ IH
.
∀x∈V(Y∗ ) ∃y ∈ k ψ k M(v|x)
=⇒
∀x∈V(Y∗ ) ψ `(M(v|x))
L.3.6
(∀v ψ)`(M) .
=⇒
¢
Assume θ is ∃v ψ. Then: .
∃d ∈ k ∃v ψ k M
⇐⇒ ∃d ∈
X
.
k ψ k M(v|j)
j∈V(Y ∗ ) .
⇐⇒ ∃j ∈ V(Y∗ ) ∃s ∈ k ψ k M(v|j) IH
=⇒ =⇒
∃j ∈V(Y∗ )ψ `(M(v|j)) (∃v ψ)`(M) . 2
15
Theorem 5.3 (CZF + REA + ΠΣW−AC) ∗ For every θ ∈ L∈ and any V(Yw )-assignment M, if θ is CC, then .
∃i∈k θ k M
⇒
θ`w (M) ,
where θ`w (M) denotes the result of replacing each free variable a of θ by `w (aM ) and each unbounded quantifier ∗ Qx of θ by Qx∈H(Yw ). Proof. This is the same proof as for the previous one.
5.1
2
“Mathematical” formulas
The previous theorem provides a collection of formulas for which inhabitedness of their formulae-as-classes interpretation implies their truth. However, it is not clear whether this collection includes many statements of workaday mathematics. To show the richness of CC, we shall coin the notion of a “mathematical” formula. Definition 5.4 The mathematical set terms are a collection of class terms inductively defined by the following clauses: 1. ω is a mathematical set term. 2. If S and T are mathematical set terms then so are [ S := {u : ∃x ∈ S u ∈ x}, {S, T }
:= {u : u = S ∨ u = T }.
3. If S and T are mathematical set terms then so are S+T S×T S→T
:= {h0, xi : x ∈ S} ∪ {h1, xi : x ∈ T }, := {hx, yi : x ∈ S ∧ y ∈ T }, := {f : f : S → T }.
4. If S, T1 , . . . , Tn are mathematical set terms and ψ(x, y1 , . . . , yn ) is a restricted formula (of set theory) then {x ∈ S : ψ(x, T1 , . . . , Tn )} is a mathematical set term. 5. If S, T1 , . . . , Tn , P1 , . . . , Pk are mathematical set terms and ψ(x, y1 , . . . , yn , z1 , . . . , zk ) is a bounded formula (of set theory) then {u : u = {x ∈ S : ψ(x, y1 , . . . , yn , P~ )} ∧ y1 ∈ T1 ∧ . . . ∧ yn ∈ Tn )} is a mathematical set term, where P~ = P1 , . . . , Pk . The generalized mathematical set terms are defined by the clauses for mathematical set terms plus the following clauses: 6. If T is a generalized mathematical set term then so is H(T ), where H(T ) denotes the smallest class Y such that ran(f ) ∈ Y whenever a ∈ T and f : a → Y . 7. If S and T are generalized mathematical set terms, then so is Wx∈S Tx . 8. If S and T are generalized mathematical set terms, then so is WF(S, T ). Here WF(S, T ) denotes the smallest class Z such that whenever a∈S and Ta = {x∈S | hx, ai ∈ T } ⊆ Z then a∈Z. 16
A mathematical formula (generalized mathematical formula) is a formula of the form ψ(T1 , . . . , Tn ), where ψ(x1 , . . . , xn ) is bounded and T1 , . . . , Tn are mathematical set terms (generalized mathematical set terms) (with the proviso that none of the free variables occurring in the Ti ’s is a bound variable of ψ). A mathematical sentence (generalized mathematical sentence) is a mathematical formula (generalized mathematical formula) without free variables. Remark 5.5 1. From the point of view of ZFC, the mathematical set terms denote sets of rank < ω + ω in the cumulative hierarchy while the generalized mathematical set terms denote sets of rank < ℵω . 2. The idea behind mathematical set terms is that they comprise all sets that one is interested in in ordinary mathematics. E.g., with the help of Definition 5.4, clauses (1) and (3) one constructs the set of natural numbers, integers, rationals, and the function space N → Q. Using clause (4) one obtains the set of Cauchy sequences of rationals from N → Q. The main application of clause (5) is made in constructing quotients. If S and R ⊆ S × S are set terms and R is an equivalence relation on S, then (5) permits one to form the set term S/R
=
{[a]R | a∈S},
where [a]R = {x∈S | hx, ai ∈ R}. Therefore, by employing clause (5), one can define the set of equivalence classes of Cauchy sequences, i.e., the set of reals. 3. Definition 5.4 clause (5) is related to the abstraction axiom of Friedman’s system B in [11]. Lemma 5.6
1. (CZF) Every mathematical set term is a set.
2. (CZF + REA) Every generalized mathematical set term is a set. Proof : We proceed by induction on the clauses for the definition of mathematical set terms. ω is a set by the Infinity Axiom. That the set terms generated by clause (2) are sets follows from the respective inductive hypothesis via the Pairing and Union Axioms. If the set terms are generated according to clause (3), one applies the respective inductive hypothesis and the fact that CZF proves the existence of the disjoint union, cartesian product, and function space of any two sets. For set terms generated according to clause (4) one uses the inductive hypothesis for the set terms S, T1 , . . . , Tn and Bounded Separation. Next, we address clause (5). By the inductive hypotheses, P~ , T~ , S are sets. Hence, using Bounded Separation, {x ∈ S : ψ(x, ~y , P~ )} is a set for every ~y ∈ T1 × · · · × Tn . Using the Replacement Schema (which is provable in CZF), {u : u = {x ∈ S : ψ(x, y1 , . . . , yn , P~ )} ∧ y1 ∈ T1 ∧ . . . ∧ yn ∈ Tn )} is a set. To prove that every generalized mathematical set term is a set on the basis of CZF + REA, we have to consider clauses (6)-(8) as well. Here we invoke [3], Corollary 5.3, namely that CZF + REA proves that H(T ), Wx∈S Tx and WF(S, T ) are sets whenever S and T are sets. u t Formally, we shall conceive of mathematical formulas and generalized mathematical formulas as defined in a certain extension Lmath of the language L∈ , namely, an extension by class terms. Strictly speaking, the formulae-as-classes interpretation is defined for formulas of L∈ only. In order to talk about the interpretation ¡ ¢♦ of L∈ -formulas, we shall fix a translation · from Lmath to L∈ . The definition below is inductive and follows the intended meaning of generalized mathematical set terms in Definition 5.4. ¡ ¢♦ for set terms S by recursion on the build-up of S: Definition 5.7 We first define x = S ¡
¢♦ x=ω ³ [ ´♦ x= S ¡ ¢♦ x = {S, T }
:= := :=
£ ¡ ¢¤ ∀u u∈x ↔ 0 = u ∨ ∃v ∈x(u = v ∪ {v}) ¤ £¡ ¢♦ ∧ ∀u(u ∈ x ↔ ∃v ∈ z u ∈ v) ∃z z = S £¡ ¢♦ ¡ ¢♦ ¤ ∃z∃w z = S ∧ w=T ∧ ∀u(u ∈ x ↔ u = z ∨ u = w) 17
¡ ¢♦ x=S+T ¡ ¢♦ x=S×T ¡ ¢♦ x=S→T
:=
∃z∃w
:=
∃z∃w
:=
∃z∃w
£¡ £¡ £¡
z=S z=S z=S
¢♦ ¢♦ ¢♦
¡ ¢♦ ¤ ∧ w=T ∧ ∀u(u ∈ x ↔ ∃v ∈z u = h0, vi ∨ ∃y ∈w u = h1, yi) ¡ ¢♦ ¤ ∧ w=T ∧ ∀u(u ∈ x ↔ ∃x∈z∃y ∈w u = hx, yi) ¡ ¢♦ ¤ ∧ w=T ∧ ∀f [f ∈ x ↔ Fun(f ) ∧ dom(f ) = z ∧ ∀y ∈z(f (y) ∈ w)] .
If Q is a set term of the form {v ∈ S : ψ(v, T~ )} then1 ³ ´♦ ¡ ¢♦ £¡ ¢♦ ¤ x=Q := ∃z∃w ~ z=S ∧ w ~ = T~ ∧ ∀v(v ∈ x ↔ v ∈ z ∧ ψ(v, w)) ~ . ¡ ¢♦ If Q is a set term of the form {u : u = {v ∈ S : ψ(v, ~y , P~ )} ∧ ~y ∈ T~ )}, then x = Q is the formula ∃z∃w ~ ∃~y
³ ´♦ ³ ´♦ £¡ ¢♦ ¤ z=S ∧ w ~ = T~ ∧ ~y = P~ ∧ ∀u(u ∈ x ↔ ∃~v ∈ w ~ u = {p ∈ z : ψ(p, ~v , ~y )}) ,
where u = {p ∈ z : ψ(p, ~v , , ~y )} stands for ∀q ∈u[q ∈ z ∧ ψ(q, ~v , ~y )] ∧ ∀q ∈z[ψ(q, ~v , ~y ) → q ∈ u]. In the case of generalized mathematical set terms we have to consider three more cases. Suppose Q is of the form H(T ), where T is a generalized mathematical set term. Put £ ¤ ψH (a, b) := ∀f ∀u∈a Fun(f ) ∧ dom(f ) = u ∧ ran(f ) ⊆ b → ∃z ∈ b [z = ran(f )] , ¡ ¢♦ £¡ ¢♦ ¤ x = H(T ) := ∃z z = T ∧ ψH (z, x) ∧ ∀w [ψH (z, w) → x ⊆ w] . Suppose Q is of the form Wx∈S Tx , where S and T are generalized mathematical set terms. Put £ ¤ ψW (a, b, c) := ∀f ∀u∈a Fun(f ) ∧ dom(f ) = bu ∧ ran(f ) ⊆ c → hu, f i ∈ c , ¡ ¢♦ £¡ ¢♦ ¡ ¢♦ ¤ x = Wu∈S Tu := ∃z ∃v z = S ∧ v=T ∧ ψW (z, v, x) ∧ ∀w [ψW (z, v, w) → x ⊆ w] . Suppose Q is of the form WF(S, R), where S and R are generalized mathematical set terms. Put £ ¤ ψWF (a, r, c) := ∀u∈a ∀v(hv, ui ∈ r → v ∈ c) → u ∈ c , ¡ ¢♦ £¡ ¢♦ ¡ ¢♦ ¤ x = WF(S, R) := ∃z ∃r z = S ∧ r=R ∧ ψWF (z, r, x) ∧ ∀w [ψWF (z, r, w) → x ⊆ w] . An arbitrary mathematical formula (generalized mathematical formula) is of the form ψ(T1 , . . . , Tn ), where T1 , . . . , Tn are mathematical set terms (generalized mathematical set terms) and ψ(z1 , . . . , zn ) is a bounded formula of L∈ . We then put ´♦ ¡ ¢♦ £³ ¤ ψ(T1 , . . . , Tn ) := ∃z1 . . . ∃zn ~z = T~ ∧ ψ(z1 , . . . , zn ) . The reason for bothering the reader with a detailed translation of mathematical formulas into the official language of set theory is that an inspection of it readily yields the following result. Lemma 5.8 If θ is a mathematical formula then θ♦ belongs to the CC formulas. This leads to the following corollaries of Theorem 5.2 and Theorem 5.3. Theorem 5.9 (CZF + ΠΣ−AC) For every mathematical formula θ and V(Y∗ )-assignment M, .
∃i∈k θ♦ k M
implies
¡ ♦ ¢`(M) , θ
where for a formula ψ, ψ `(M) denotes the result of replacing each free variable a of ψ by `(aM ) and each unbounded quantifier Qx of ψ by Qx∈H(Y∗ ). a vector of set terms T~ ≡ T1 , . . . , Tn we write ~ y ∈ T~ and ¡ ¢♦ . . . ∧ yn ∈ Tn , respectively. 1 For
18
³ ´♦ ¡ ¢♦ ~ y = T~ for y1 ∈ T1 ∧ . . . ∧ yn ∈ Tn and y1 ∈ T1 ∧
Theorem 5.10 (CZF + REA + ΠΣW−AC) ∗ For every mathematical formula θ and a V(Yw )-assignment M .
∃i∈k θ♦ k M
implies
¡ ♦ ¢`w (M) θ ,
where for a formula ψ, ψ `w (M) denotes the result of replacing each free variable a of ψ by `w (aM ) and each ∗ unbounded quantifier Qx of ψ by Qx∈H(Yw ). We would like to expand the previous result to generalized mathematical formulas, the obstacle being that these formulas need not be in CC. Definition 5.11 A class A is regular if it is transitive and for every a ∈ A and set R ⊆ a × A, if ∀x ∈ a ∃y hx, yi ∈ R then there is a set b ∈ A such that ∀x ∈ a ∃y ∈ b hx, yi ∈ R ∧ ∀y ∈ b ∃x ∈ a hx, yi ∈ R. Definition 5.12 Let ΠΣ−PAx be the assertion that every ΠΣ-generated set is a base and every set is an image of a ΠΣ-generated set. Similarly, one defines ΠΣW−PAx. Lemma 5.13
1. (CZF + ΠΣ−AC) H(Y∗ ) is a regular model of CZF + DC + ΠΣ−AC + ΠΣ−PAx.
∗ 2. (CZF + REA + ΠΣW−AC) H(Yw ) is a regular model of CZF + REA + DC + ΠΣ−AC + ΠΣW−PAx. ∗ ). (1) then follows from [3], Theorem Proof : By Lemma 3.4, we have H(Y) = H(Y∗ ) and H(Yw ) = H(Yw 4.2 and (2) follows from [3], Theorem 5.10. 2 ∗ Definition 5.14 If θ is a generalized mathematical formula with parameters in V(Yw ) we shall use the abbreviation ¡ ¢♦ ∗ V(Yw ) |= θ := ∃i ∈ k θ k M . ∗ Likewise, if θ is a generalized mathematical formula with parameters in H(Yw ) we shall use the abbreviation ∗ H(Yw ) |= θ
¡ ¢♦ ∗ ∗ iff θ holds in H(Yw ), i.e. with all unbounded quantifiers restricted to H(Yw ). ∗ Lemma 5.15 (CZF + REA + ΠΣW−AC) Let α, β, γ ∈ V(Yw ) and α˙ = `w (α), β˙ = `w (β), and γ˙ = `w (γ). Then we have the following. ∗ ∗ V(Yw ) |= β = H(α) ⇒ H(Yw ) |= β˙ = H(α). ˙ ∗ ∗ V(Yw ) |= γ = Wu∈α βu ⇒ H(Yw ) |= γ˙ = Wu∈α˙ β˙ u . ∗ ∗ ˙ V(Yw ) |= γ = WF(α, β) ⇒ H(Yw ) |= γ˙ = WF(α, ˙ β).
(16) (17) (18)
∗ Proof : Assume V(Yw ) |= β = H(α). The formula ψH (α, β) of Definition 5.7 is a formula which starts with a universal quantifier and is followed by a bounded matrix, and thus, by Theorem 5.3, ∗ ˙ H(Yw ) |= ψH (α, ˙ β).
(19)
∗ ∗ ∗ Since H(Yw ) is a model of CZF + REA by Lemma 5.13, there exists b ∈ H(Yw ) such that H(Yw ) |= ∗ ∗ ˙ b = H(α). ˙ As `w is surjective there exists ρ ∈ V(Yw ) such that ρ˙ = b. From (19) we deduce H(Yw ) |= ρ˙ ⊆ β, and hence, using Theorem 3.15, ∗ V(Yw ) |= ρ ⊆ β.
19
(20)
∗ Next we would like to show that also V(Yw ) |= β ⊆ ρ. Here we have to resort to a different description of H(α). By Lemma 2.7, we have that provably in CZF,
x ∈ H(α)
⇔ ∃G∃u[G is good ∧ x ∈ Ga ],
(21)
where “ G is good” stands for £ ¤ ∀ hv, yi ∈ G ∃b ∈ α ∃f Fun(f ) ∧ f : b → G∈v ∧ ran(f ) = y] . Letting ψg (α, x) denote the formula on the right hand side of (21), we see that ψg (α, x) belongs to CC. ∗ ∗ ) is a model of CZF by Theorem 4.14, we can employ the foregoing ) |= η ∈ β. As V(Yw Now suppose V(Yw ∗ considerations to express this fact via the CC formula ψg (α, η), so that V(Yw ) |= ψg (α, η) and therefore, by ∗ ∗ ∗ Theorem 5.3, H(Yw ) |= ψg (α, ˙ η). ˙ As H(Yw ) is a model of CZF as well we arrive at H(Yw ) |= η˙ ∈ H(α). ˙ ∗ ∗ ∗ Hence H(Yw ) |= η˙ ∈ ρ˙ and so (by Theorem 3.15) V(Yw ) |= η ∈ ρ, showing that V(Yw ) |= β ⊆ ρ. Thus, in ∗ conjunction with (20), we get V(Yw ) |= β = ρ, yielding ∗ ) |= β˙ = H(α). ˙ H(Yw
The proofs of the other cases are similar and utilize the same considerations.
u t
Theorem 5.16 (CZF + REA + ΠΣW−AC) ∗ For every generalized mathematical formula θ and V(Yw )-assignment M, .
∃i∈k θ♦ k M
implies
¡ ♦ ¢`w (M) θ ,
where for a formula ψ, ψ `w (M) denotes the result of replacing each parameter a of ψ by `w (aM ) and each ∗ unbounded quantifier Qx of ψ by Qx∈H(Yw ). ´♦ £³ ¤ Proof. θ♦ is of the form ∃z1 . . . ∃zn ~z = T~ ∧ ψ(z1 , . . . , zn ) , where ψ(z1 , . . . , zn ) is a bounded formula and the T~ are generalized set terms. The assertion then follows from Lemma 5.15 taken together with Theorem 3.15. 2
5.2
Absoluteness of mathematical formulas
In this subsection we show that mathematical formulas are absolute for H(Y∗ ) and that generalized math∗ ematical are absolute for H(Yw ). Lemma 5.17 (CZF + ΠΣ−AC) Let S be a set term with parameters in H(Y∗ ). By 5.13, H(Y∗ ) is a ∗ model of CZF, and thus S is interpreted as a set in H(Y∗ ). Let S H(Y ) be the interpretation of S in H(Y∗ ). H(Y ∗ ) Then S = S . Proof : The proof proceeds by induction on the generation of S. Note that except for the case when S is of the form T → P , this is obvious because of the absoluteness of bounded formulas. ∗ Suppose S is of the form T → P . From the inductive hypotheses for T and P we get T = T H(Y ) ∗ and P = P H(Y ) , in particular T, P ∈ H(Y∗ ). Since H(Y∗ ) is a model of CZF it suffices to show that ∗ (T → P ) ⊆ H(Y∗ ) to be able to conclude that (T → P ) = (T → P )H(Y ) . Let f : T → P . Since T ∈ H(Y∗ ) there exists A ∈ Y∗ and g : A → T such that T = ran(g). Now define h : A → H(Y∗ ) by h(i)
= hg(i), f (g(i))i.
Then ran(h) ∈ H(Y∗ ) and, moreover, ran(h) = f , whence f ∈ H(Y∗ ).
2
∗ Lemma 5.18 (CZF + REA + ΠΣW−AC) Let S be a generalized set term with parameters in H(Yw ). ∗ ∗ ∗ By 5.13, H(Yw ) is a model of CZF + REA, and thus S is interpreted as a set in H(Yw ). Let S H(Yw ) be ∗ ∗ the interpretation of S in H(Yw ). Then S = S H(Yw ) .
20
Proof : Again, the proof proceeds by induction on the generation of S. In addition to the cases of the previous lemma, we have to consider inductively defined set terms. Suppose S = H(T ). By the inductive ∗ assumption we then have T H(Yw ) = T . We will call a set of ordered pairs G good if ∀ ha, yi ∈ G ∃f ∃b∈T [f : b → G∈a ∧ y = ran(f )],
S where G∈a = b∈a Gb and Gb = {u | hb, ui ∈ G}. By Lemma 2.7 we get ∗
x ∈ (H(T ))H(Yw )
iff
∗ H(Yw ) |= ∃G ∃a [G is good ∧ x ∈ Ga ].
As the property of being good is formalizable by a Σ formula and therefore upward persistent, x ∈ ∗ (H(T ))H(Yw ) implies ∃G ∃a [G is good ∧ x ∈ Ga ], and thence, by Lemma 2.7, x ∈ H(T ). In consequence, ∗ ∗ H(Yw ) (H(T )) ⊆ H(T ). To establish the converse inclusion, suppose c ∈ T and f : c → (H(T ))H(Yw ) . In ∗ H(Yw ) the course of the proof of Lemma 5.17 it was shown that the latter yields f ∈ (H(T )) , hence we get ∗ ∗ ran(f ) ∈ (H(T ))H(Yw ) . Having shown that (H(T ))H(Yw ) is closed under the clauses defining H(T ), we ∗ conclude H(T ) ⊆ (H(T ))H(Yw ) . The cases where S = Wx∈P Tx or S = WF(P, R) are dealt with in the same way as in the case of S = H(T ). 2 Definition 5.19 Let Σ(math) (Σ(gmath)) denote the smallest collection of formulas which comprises the mathematical set formulas (the generalized mathematical set formulas) and is closed under ∧, ∨, bounded quantification, and unbounded existential quantification. Corollary 5.20 (i) (CZF + ΠΣ−AC) Let ψ be a Σ(math) formula with parameters in H(Y∗ ). If H(Y∗ ) |= ψ, then ψ. ∗ ∗ (ii) (CZF + REA + ΠΣ−AC) Let ψ be a Σ(gmath) formula with parameters in H(Yw ). If H(Yw ) |= ψ, then ψ.
Proof : This follows readily by induction on ψ using Lemma 5.17 and Lemma 5.18, respectively. Theorem 5.21
2
∗
(i) (CZF + ΠΣ−AC) Let θ be a Σ(math) sentence. If V(Y ) |= ψ, then ψ holds true.
∗ (ii) (CZF + REA + ΠΣW−AC) Let θ be a Σ(gmath) sentence. If V(Yw ) |= ψ, then ψ holds true.
Proof : (i) is a consequence of Theorem 5.9 and Corollary 5.20, (i), while (ii) follows from Theorem 5.16 in conjunction with Corollary 5.20, (ii). u t [2, 3] feature several more choice principles. The main reason for their omission is that these axioms have no impact on the preceding result. This will be made precise below. Definition 5.22 Let BCAΠ be the statement that whenever A is a base and Ba is a base for each a ∈ A, Q then x∈A Bx is a base. Let BCAI be the statement that whenever A is a base then I(A, b, c) is a base for all b, c ∈ A. Theorem 5.23 Let ψ be a mathematical sentence and let θ be a generalized mathematical sentence. Then the following hold: (i) CZF + ΠΣ−AC ` ψ if and only if CZF + ΠΣ−AC + ΠΣ−PAx + BCAΠ + BCAI + RDC ` ψ. (ii) CZF + REA + ΠΣW−AC ` θ if and only if CZF + REA + ΠΣW−PAx + BCAΠ + BCAI + RDC ` θ. Proof : (i): Arguing in CZF + ΠΣ−AC one can show that H(Y∗ ) is a model of CZF + ΠΣ−AC + RDC + ΠΣ−AC + ΠΣ−PAx by the same proof as for [3], Theorem 4.2. By Corollary 2.12, ΠΣ−AC and ΠΣI−AC are equivalent over CZF, and ΠΣI−AC implies BCAΠ and BCAI . To see this note that by 4.8 of [2] the class of bases is the class of those sets that are in one-one correspondence with a ΠΣI-generated set from which it follows that the class of bases is ΠΣI-closed and hence BCAΠ and BCAI hold. The upshot is that H(Y∗ ) is also a model of BCAΠ and BCAI . Hence (i) follows owing to Corollary 5.20 (i). (ii) is proved similarly, this time by utilizing Corollary 5.20 (ii) and [3] Theorem 5.10.
21
u t
6
Interpretations of type theory in CZF and CZF + REA
In the series of papers [1, 2, 3], Aczel gave interpretations of CZF and CZF + REA in Martin-L¨of’s intuitionistic type theory. By ruminating on these interpretations one can delineate particular systems ML1V and ML1WV of type theory that are sufficient unto this task. In what follows we assume familiarity with type theory as presented in Martin-L¨of’s 1984 monograph [14] or in Beeson’s book [7]. The basic system of type theory, notated by ML0 , is the one with the type constructors N,N0 , N1 , Π, Σ, +, I. In [13, 14] Martin-L¨ of considered an infinite, externally indexed tower of universes U1 ∈ U2 ∈ . . . ∈ Un ∈ . . . all of which are closed under the standard ensemble of type forming operations. By ML1 we shall denote the extension of ML0 by one universe U plus rules to the effect that U is closed under the above constructors. ML1W denotes the extension of ML1 wherein the universe U is also closed under taking W-types (see 6.1 below). The formalisation of universes for intuitionistic type theory we use in this section is that referred to as the Russell formulation in Martin-L¨of’s monograph [14]. The fundamental notions of type theory are introduced in the four forms of judgement: A is a type (abbr. A type), A and B are equal types (abbr. A = B), a is an element of type A (abbr. a : A), and a, b are equal elements of type A (abbr. a = b : A). We prefer to use the colon “ :” rather than the elementhood symbol “ ∈” to stress the distinction between set theory and type theory. The rule of type theory are presented in natural deduction style as in [14]. The judgements within brackets indicate discharged assumptions. Definition 6.1 The introduction rules of ML1W concerning the W-type are the following: [x : A] A : U F (x) : U W(A, F ) : U W(A, F ) : U W(A, F ) type. Combining the foregoing rules gives rise to the derived rule of restricted W–formation, [x : A] A : U F (x) : U . W(A, F ) type
(res–W–formation)
Definition 6.2 The theories ML1V and ML1WV are obtained from ML1 and ML1W , respectively, by equipping them with Aczel’s type of iterative sets V (cf. [1]). The rules pertaining to V are: (V–formation) V type (V–introduction)
(V–elimination)
(V–equality)
A : U f : A→V sup(A, f ) : V
[A : U, f : A → V] [z : (Πv : A)C(f (v))] c : V d(A, f, z) : C(sup(A, f )) TV (c, (A, f, z)d) : C(c)
[A : U, f : A → V] [z : (Πv : A)C(f (v))] B : U g : B→V d(A, f, z) : C(sup(A, f )) . TV (sup(B, g), (A, f, z)d) = d(B, g, (λv)TV (g(v), (A, f, z)d)) : C(sup(B, g))
In order to define their interpretations in set theory, we need a detailed account of the syntax of ML1V and ML1WV. Here we will follow [7, Ch. XI]; however, for the readers convenience, we shall recall most of the definitions. If B is any expression, and x1 , . . . , xn are variables, we form the expression (x1 , . . . , xn )B. The 4 symbol ≡ will be used for the relation on expressions satisfying 4
((x1 , . . . , xn )B)(x1 , . . . , xn ) ≡ B 4
and A ≡ C for expressions A and C which differ only in the renaming of bound variables (cf. [7, XI6]). 22
Definition 6.3 (cf. [7, XI.20.3]). The constants of ML1 V are: Π, Σ, I, +, N, 0, sN , r, λ, ap, sup, E, i, j, D, J, R, TV , U, V and for each natural number m, Nm and Rm . ML1WV also has the constant W. The terms are generated by: 1. Every constant and variable is a term; 2. If t and s are terms, then t(s) and (t, s) are terms; 3. If t is a term, then (x1 , . . . , xn )t is a term, where the xi are variables. Free and bound occurrences of variables in terms are defined as usual, letting abstraction, i.e. the formation of (x1 , . . . , xn )t, bind the variables x1 , . . . , xn . We now would like to assign to every term t of ML1 V a corresponding application term t∗ by replacing the abstract application of ML1 V with set-recursive application. It is then a straightforward matter to translate a formula of the form t∗ : X into a legitimate formula of CZF. Definition 6.4 We now assign to each term t of ML1 V an application term t∗ . Occurrences of λ in the definition of t∗ denote the λ-operator introduced by Lemma 4.6. We fix two new natural numbers u ¯ and v ¯. We shall write (x, y) for {p}(x, y) and, inductively, (x1 , . . . , xk+1 ) for {p}((x1 , . . . , xk ), xk+1 ). For constants c we define c∗ by: 0∗ is 0 Π∗ is λxλy.πxy Σ∗ is λxλy.σxy +∗ is λxλy.plxy I∗ is λzλxλy.izxy N∗ is ω N∗k is k¯ U∗ is u ¯ V∗ is v ¯ s∗N is sN r∗ is 0 λ∗ is λx.x (i.e. skk) ap∗ is λxλy.yx sup∗ is λxλy.(x, y) E∗ is λxλy.y(p1 x, p1 x) i∗ is λx.(0, x) j∗ is λx.(1, x) D∗ is λxλyλz.(0, p1 x, y(p1 x), z(p1 x)) J∗ is λxλy.y. R∗k is λm.λx0 . . . λxk−1 .ek [m, x0 , . . . , xk−1 ], where an application term ek is chosen so that CZF proves (ek [m, x0 , . . . , xk−1 ] ' xm )∧ if m < k. R∗ is an application term introduced by the Recursion Theorem 4.7 to satisfy (R∗ ab0 ' a)∧ and (R∗ ab(sN x) ' bx(R∗ abx))∧ . T∗V = T∗V [e] is a term introduced by the Recursion Theorem ({T∗V }(sup∗ (a, b), λx.λy.λz.{e}(x, y, z)) ' {e}(a, b, (λx.x)(λv.{T∗V }(ap∗ (b, v), λx.λy.λz.{e}(x, y, z)))))∧ . For a variable u let u∗ be u. For complex terms of ML1 V we define: ((x1 , . . . , xn )t)∗ is λx1 . . . λxn .t∗ ; (t(s))∗ is t∗ s∗ ; (t, s)∗ is {p}(t∗ , s∗ ). Definition 6.5 The type terms of ML1 V are defined inductively by 1. N and Nk are type terms (for each integer k); 2. If A and B are type terms, so is (A+B); 23
3. If B(x) and A are type terms, and x is not free in A or in B, then Π(A, B) and Σ(A, B) are type terms; 4. If A is a type term and t, s are any terms of ML1 V, then I(A, s, t) is a type term; 5. U and V are type terms; 4
6. If A is a type term and B ≡ A, then B is a type term. Definition 6.6 Large types (type terms) are those containing the constants U or V. Others are small types. Definition 6.7 (Interpretation of ML1 V in CZF) By induction on the complexity of the type term A we shall assign to each judgement Φ of ML1 V of the form u : A or u = v : A (u, v variables) a formula (Φ)∧ of CZF with the same free variables. (ux : A)∧ and (ux = uy : A)∧ will be used as shorthand for ∃z[{u}(x) = z ∧ (z : A)∧ ] and ∃z[{u}(x) = z ∧ {u}(y) = z ∧ (z : A)∧ ], respectively. Likewise, (u : A(vx))∧ abbreviates ∃z[{v}(x) = z ∧ (u : A(z))∧ ], etc. Also, (ux : A)∧ and (ux = uy : A)∧ will be used as shorthand for ∃z[{u}(x) = z ∧ (z : A)∧ ] and ∃z[{u}(x) = z ∧ {u}(y) = z ∧ (z : A)∧ ], respectively. The clauses in the definition are as follows: (u : Π(A, B))∧
is
(u : Π(A, B))∧
is
(u = v : Π(A, B))∧
is
(u = v : Π(A, B))∧
is
(u : Σ(A, B))∧ (u = v : Σ(A, B))∧
is is
(u : (A+B))∧ (u = v : (A+B))∧
is is
(u : I(A, b, c))∧ (u = v : I(A, b, c))∧ (u : N)∧ (u = v : N)∧
is is is is
(u : Nk )∧ (u = v : Nk )∧
is is
(u : U) (u = v : U)∧
is is
∧
Fun[u] ∧ ∀z ∈u(((z)0 : A)∧ ∧ ((z)1 : B((z)0 ))∧ ) ∧ ∀x[(x : A)∧ → (ux : B(x))∧ ] ∧ ∀x, y[(x = y : A)∧ → (ux = uy : B(x))∧ ] if A is a small type ∀x[(x : A)∧ → (ux : B(x))∧ ] ∧ ∀x, y[(x = y : A)∧ → (ux = uy : B(x))∧ ] if A is a large type (u : Π(A, B))∧ ∧ (v : Π(A, B))∧ ∧ ∀x[(x : A)∧ → (ux = vx : B(x))∧ ] if A is a small type (u : Π(A, B))∧ ∧ (v : Π(A, B))∧ ∧ ∀x[(x : A)∧ → (ux = vx : B(x))∧ ] if A is a large type u = h(u)0 , (u)1 i ∧ ((u)0 : A)∧ ∧ ((u)1 : B((u)0 ))∧ (u : Σ(A, B))∧ ∧ (v : Σ(A, B))∧ ∧ ((u)0 = (v)0 : A)∧ ∧ ((u)1 = (v)1 : B((u)0 ))∧ ¡ ¢ u = h(u)0 , (u)1 i ∧ [(u)0 = 0 ∧ ((u)1 : A)∧ ] ∨ [(u)0 = 1 ∧ ((u)1 : B)∧ ] (u : (A+B))∧ ∧ (v : (A+B))∧ ∧ ¡ [(u)0 = 0 ∧ (v)0 = 0 ∧ ((u)1 = (v)1 : A)∧ ] ∨ ¢ [(u)0 = 1 ∧ (v)0 = 1 ∧ ((u)1 = (v)1 : B)∧ ] u = 0 ∧ (b = c : A)∧ u = 0 ∧ v = 0 ∧ (b = c : A)∧ u∈ω u=v∧u∈ω u ∈ k¯ u = v ∧ u ∈ k¯ u ∈ Y∗ u = v ∧ u ∈ Y∗ ∧ v ∈ Y∗ 24
(u : V)∧ (u = v ∈ V)∧
is is
u ∈ V(Y∗ ) u = v ∧ u ∈ V(Y∗ ) ∧ v ∈ V(Y∗ )
If s and t are arbitrary terms of ML1 V and A is a type term of ML1 V, we set: (t : A)∧ (s = t : A)∧
is is
∃u[(t∗ ' u)∧ ∧ (u : A)∧ ], ∃u, v[(s∗ ' u)∧ ∧ (t∗ ' v)∧ ∧ (u = v : A)∧ ].
For type terms A and B we define (A = B)∧ by ∀u[(u : A)∧ ↔ (u : B)∧ ] ∧ ∀u, v[(u = v : A)∧ ↔ (u = v : B)∧ ]. In the natural deduction style presentation of type theory one deduces hypothetical judgements, i.e. judgements which are made under assumptions (see [14] pp. 16–20). We shall use the notation ML1V ` Φ(u1 , . . . , un ) [u1 : A1 , . . . , un : A(u1 , . . . , un−1 )] to convey that the judgement Φ(u1 , . . . , un ) is deducible in ML1V under the open assumptions u1 : A1 , . . . , un : A(u1 , . . . , un−1 ). Theorem 6.8 (Soundness of the Interpretation of ML1 V in CZF) If ML1V ` Φ(u1 , . . . , un ) [u1 : A1 , . . . , un : A(u1 , . . . , un−1 )], where Φ(u1 , . . . , un ) is a judgement not of the form “A type”, then CZF ` (u1 : A1 )∧ ∧ . . . ∧ (un : A(u1 , . . . , un−1 ))∧ → (Φ(u1 , . . . , un ))∧ . Proof. First note that if an expression of the form “A type”, s : A, s = t : A, or A = B appears in a derivation of ML1 V, then A is a type term in the sense of Definition 6.5, as is readily seen by induction on derivations in ML1 V. This ensures that any judgment of ML1 V gets translated under ∧ . Secondly, it should be clear that the above interpretation replaces the abstract application of ML1 V by set-recursive application in a faithful way, i.e. the equations which the rules of ML1 V prescribe for the constants of ML1 V are satisfied by their translations. The constructions 2.8 and 3.1 ensure that particular rules for U and V-introduction are sound with respect to the interpretation ∧ . The soundness of V-elimination and V-equality is verified in the same way as in [19, Th.4.11]. u t The foregoing interpretation can be extended to ML1WV. ML1WV has the additional constants W and TW , where TW is the eliminatory constant associated with the W-type. ML1WV has additional type terms of the form W(A, B) providing A is a small type and B(x) is a small type for every x : A. Here a small type is one that does not involve U or V (but may contain W). The translation of 6.4 has to be altered for the types U and V as follows (u : U)∧ (u = v : U)∧ (u : V)∧ (u = v ∈ V)∧
is is is is
∗ u ∈ Yw ∗ ∗ ∧ v ∈ Yw u = v ∧ u ∈ Yw ∗ u ∈ V(Yw ) ∗ ∗ u = v ∧ u ∈ V(Yw ) ∧ v ∈ V(Yw )
¯ and is to be continued to terms of ML1WV by letting W∗ be λxλy.wxy and T∗W be defined similar to T∗V in 6.4. Next, building on 6.7, we need to translate judgements of the form u : W(A, B) and u = v : W(A, B). (u : W(A, B))∧
is
Wz∈A∗ B ∗ (z),
where A∗ = {x | (x : A)∧ } and the function B ∗ with domain A∗ is defined by B ∗ (x) = {z | (z : B(x))∧ }. (u = v : W(A, B))∧
is
(u : W(A, B))∧ ∧ (v : W(A, B))∧ ∧ ((u)0 = (v)0 : A)∧ ∧ ∀x[(x : A)∧ → ((u)1 (x) = (v)1 (x) : B(x))∧ ].
The interpretation of ML1V in CZF given in 6.8 can then be extended as follows. Theorem 6.9 (Soundness of the Interpretation of ML1WV in CZF + REA) If ML1WV ` Φ(u1 , . . . , un ) [u1 : A1 , . . . , un : A(u1 , . . . , un−1 )], where Φ(u1 , . . . , un ) is a judgement not of the form “A type”, then CZF + REA ` (u1 : A1 )∧ ∧ . . . ∧ (un : A(u1 , . . . , un−1 ))∧ → (Φ(u1 , . . . , un ))∧ . 25
7
Putting it together
Definition 7.1 By V-assignment we mean a function M : Var → V. Definition 7.2 (See [7, XII.1.4].) For every formula ϕ of CZF and V-assignment M, we define a type term kϕkM of ML1 V. First, for α, β ∈ V, recall that kα = βk is defined by recursion theorem to satisfy ¯ λy.k˜ ˜ ¯ λx.Σ(¯ ˜ kα = βk = Π(α ¯ , λx.Σ(β, α(x) = β(y)k)) × Π(β, α, λy.kβ(x) =α ˜ (y)k)). The rest of the definition is as follows: ku ∈ vkM
is
¯ λy.kα = β(y)k) ˜ Σ(β, where α = M(u), β = M(v),
kϕ0 ∧ ϕ1 kM
is
kϕ0 kM × kϕ1 kM
kϕ0 ∨ ϕ1 kM
is
kϕ0 kM + kϕ1 kM
kϕ0 ⊃ ϕ1 kM
is
kϕ0 kM → kϕ1 kM
k⊥kM
is
k∀u∈αϕkM
is
k∃u∈αϕkM
is
k∀uϕkM
is
k∃uϕkM
is
N0 ¡ ¢ Π α ¯ , λx.kϕkM(u|α(x)) ˜ ¡ ¢ Σ α, ¯ λx.kϕkM(u|α(x)) ˜ ¡ ¢ Π V, λα.kϕkM(u|α) ¡ ¢ Σ V, λα.kϕkM(u|α) .
If ϕ is a set-theoretic formula whose free variables are among u1 , . . . , un , α1 , . . . , αn : V, and the Vassignment M satisfies M(vi ) = αi for i = 1, . . . , n we also write “ kϕ(α1 , . . . , αn )k” for “ kϕkM ”. It is easy to prove by induction on the complexity of ϕ that kϕkM is a type for all formulas ϕ, and a small type for bounded ϕ. We note that, according to the constructions 2.8 and 3.1, the type U of ML1V can be identified with the class Y∗ of sets, and the type V can be identified with the class V(Y∗ ). This in particular means that Martin-L¨ of types belonging to U or V have their set-theoretic counterparts in Y∗ and V(Y∗ ). In this sense we will identify a V-assignment with a V(Y∗ )-assignment. Likewise, owing to the constructions 2.9 and 3.3, the type U of ML1WV can also be identified with the class ∗ ∗ Yw of sets, and the type V of ML1WV can be identified with the class V(Yw ). This in particular means ∗ that the small types of ML1WV have their set-theoretic counterparts in Yw . In this sense we will identify ∗ a V-assignment with a V(Yw )-assignment. Lemma 7.3 (CZF) For every set-theoretic formula ϕ whose free variables are among u1 , . . . , un and α1 , . . . , αn ∈ V(Y∗ ), (x : kϕ(α1 , . . . , αn )k)∧ implies x ∈ k ϕ(α1 , . . . , αn ) k. Proof. This follows by comparing Definitions 4.10, 7.2 and the translation 6.7.
u t
Lemma 7.4 (CZF + REA) ∗ For every set-theoretic formula ϕ whose free variables are among u1 , . . . , un and α1 , . . . , αn ∈ V(Yw ), ∧ (x : kϕ(α1 , . . . , αn )k) implies x ∈ k ϕ(α1 , . . . , αn ) k. Proof. Similar to 7.3.
u t
Theorem 7.5 If ϕ is a formula in CC with at most u1 , . . . , un free and ML1 V ` t : kϕ(α1 , . . . , αn )k [α1 : V, . . . , αn : V] for some term t, then ¡ ¢H(Y∗ ) . CZF + ΠΣ−AC ` α1 , . . . , αn ∈ V(Y∗ ) → ϕ `(α1 ), . . . , `(αn ) 26
(22)
Proof. By Theorem 6.8 we have CZF ` α1 , . . . , αn ∈ V(Y∗ ) → ∃u((t∗ ' u)∧ ∧ (u : kϕ(α1 , . . . , αn )k)∧ ). By Lemma 7.3 we get ¡ ¢ CZF ` α1 , . . . , αn ∈ V(Y∗ ) → ∃u u ∈ k ϕ(α1 , . . . , αn ) k which by Theorem 5.2 implies (22).
u t
Theorem 7.6 If ϕ is a generalized mathematical formula with at most u1 , . . . , un free and ML1WV ` t : kϕ(α1 , . . . , αn )k [α1 : V, . . . , αn : V] for some term t, then ∗ ¡ ¢H(Yw ) ∗ CZF + REA + ΠΣW−AC ` α1 , . . . , αn ∈ V(Yw ) → ϕ `w (α1 ), . . . , `w (αn ) .
(23)
Proof. By Theorem 6.9 we have ∗ CZF + REA ` α1 , . . . , αn ∈ V(Yw ) → ∃u((t∗ ' u)∧ ∧ (u : kϕ(α1 , . . . , αn )k)∧ ).
By Lemma 7.4 we get ¡ ¢ ∗ CZF + REA ` α1 , . . . , αn ∈ V(Yw ) → ∃u u ∈ k ϕ(α1 , . . . , αn ) k which by Theorem 5.16 implies (22).
u t
Theorem 7.7 Let ψ be a mathematical sentence and let θ be a generalized mathematical sentence. Then the following hold: (i) CZF + ΠΣ−AC ` ψ if and only if ML1V ` tψ : kψk for some term tψ of ML1V . (ii) CZF + REA + ΠΣW−AC ` θ if and only if ML1WV ` tθ : kθk for some term tθ of ML1WV. Proof : The directions “ ⇒” follow by scrutinizing the proofs in [1, 2, 3]. Now suppose that ML1V ` tψ : kψk ∗ for some term tψ of ML1V. By 5.8, ψ is a CC-formula so that by 7.5 we arrive at CZF+ΠΣ−AC ` ψ H(Y ) , whence CZF + ΠΣ−AC ` ψ owing to 5.20(i). ∗ Next assume ML1WV ` tθ : kθk. Then 7.6 yields CZF + REA + ΠΣW−AC ` θH(Yw ) , and whence CZF + REA + ΠΣW−AC ` θ follows from 5.20(ii). u t
8
The existence property
It is often considered a hallmark of intuitionistic systems that they possess the disjunction and existential definability properties. Definition 8.1 Let T be a theory whose language, L(T ), encompasses the language of set theory. Moreover, for simplicity, we shall assume that L(T ) has a constant ω denoting the set of von Neumann natural numbers and for each n a constant n ¯ denoting the n-th natural number. 1. T has the disjunction property, DP, if whenever T ` ψ ∨ θ then T ` ψ or T ` θ. n) for some n. 2. T has the numerical existence property, NEP, if whenever T ` (∃x∈ω)φ(x) then T ` φ(¯ 3. T has the existence property, EP, if whenever T ` ∃xφ(x) then T ` ∃!x [ϑ(x) ∧ φ(x)] for some formula ϑ. 27
Slightly abusing terminology, we shall also say that T enjoys any of these properties if this holds only for a definitional extension of T rather than T . ZF and ZFC do not have the existence property. But even classical set theories can have the EP. Kunen observed that an extension of ZF has the EP if and only if it proves that all sets are ordinal definable, i.e., V = OD. Going back to intuitionistic set theories, let IZFR result from IZF by replacing Collection with Replacement, and let CST be Myhill’s constructive set theory of [17]. Theorem 8.2
(i) (Myhill) IZFR and CST have the DP, NEP, and the EP.
(ii) (Beeson) IZF has the DP and the NEP. (iii) (Friedman) IZF does not have the EP. (iv) (Rathjen) CZF and CZF + REA have the DP and the NEP. Proof : (i) is proved in [17]. For (ii) see [6] and for (iii) see [12]. (iv) is shown in [22].
u t
The question of whether CZF satisfies the existence property is currently unanswered. Friedman’s proof of the failure of EP for IZF seems to single out Collection as the culprit. However, that proof does not seem to carry over to CZF since the refutation of EP uses existential statements of the form £ ¤ ∃b ∀u∈a ∃y ϕ(u, y) → ∀u∈a ∃y∈b ϕ(u, y) , which are deducible in IZF by employing Collection and full Separation, but needn’t be deducible in CZF. The first author conjectures that EP fails for CZF on account of Subset Collection (and maybe Collection). There are, however, positive answers available for CZF + ΠΣ−AC and CZF + REA + ΠΣW−AC in that these theories can be shown to have the EP for mathematical and generalized mathematical statements, respectively. Theorem 8.3 Let θ1 , θ2 be Σ(math) sentences and let ψ(x) be a Σ(math) formula with at most x free. Then we have the following: (i) If CZF + RDC + ΠΣ−AC ` θ1 ∨ θ2 then CZF + ΠΣ−AC ` θ1 or CZF + ΠΣ−AC ` θ2 . (ii) If CZF + RDC + ΠΣ−AC ` ∃u ∈ ω ψ(u) then there exists a natural number n such that CZF + ΠΣ−AC ` ψ(¯ n). (iii) If CZF + RDC + ΠΣ−AC ` ∃xψ(x) then there is a formula ϑ(x) (with at most x free) such that CZF + ΠΣ−AC ` ∃!x[ϑ(x) ∧ ψ(x)]. Proof : (i): Suppose CZF + RDC + ΠΣ−AC ` θ1 ∨ θ2 . By [21], Theorem 4.14 one can £ ¤ (primitive recursively) find a closed application term t such that CZFExp ` ∃x t ' x ∧ x ∈ k θ1 ∨ θ2 k so that ¡ ¢ CZFExp ` ∃i ∈ ω [i = 0 ∧ ∃u u ∈ k θ1 k] ∨ [i = 1 ∧ ∃u u ∈ k θ2 k] . As CZFExp has the numerical existence property, the latter implies CZF ` ∃u u ∈ k θ1 k or CZF ` ∃u u ∈ k θ2 k, whence by Theorem 5.21 (i), CZF + ΠΣ−AC ` θ1 or CZF + ΠΣ−AC ` θ2 . (ii): Suppose CZF+RDC+ΠΣ−AC ` ∃u ∈ ω ψ(u). By [21], recursively) ¡ Theorem 4.14 one can (primitive ¢ find a closed application term t0 such that CZFExp ` ∃x t0 ' x ∧ x ∈ k ∃u ∈ ω ψ(u) k . At this point ∗ we have to go back to the details of the proof of [21], Lemma 4.17. The role by¢ ¡ of ω in V(Y ) is played ∗ ∗ ω = sup(ω, hω ), where hω : ω → V(Y ). We then obtain CZFExp ` ∃y t ' y ∧ y ∈ k ∃u ∈ ω ∗ ψ(u) k for a closed application term t, and thence CZFExp ` ∃i ∈ ω ∃z z ∈ k ψ(hω (i)) k. Since CZFExp enjoys the NEP, there exists a natural number n such that CZFExp ` ∃z z ∈ k ψ(hω (¯ n)) k. It also follows from the definition of hω (cf. [21],4.14) that `(hω (¯ n)) = n ¯ . Thus, by Theorem 5.21 (i), CZF + ΠΣ−AC ` ψ(¯ n). (iii): Now suppose CZF + RDC + ΠΣ−AC ` ∃xψ(x). Then, owing to [21], Theorem 4.14, one can
28
£ ¤ (primitive recursively) find a closed application term t such that CZFExp ` ∃z t ' z ∧ z ∈ k ∃xψ(x) k so that £ ¤ CZFExp ` ∃α ∈ V(Y∗ ) p0 t ' α ∧ p1 t ∈ k ψ(α) k . By 5.21 (i) the latter yields £ ¤ CZF + ΠΣ−AC ` ∃α ∈ V(Y∗ ) p0 t ' α ∧ ψ(`(α)) . £ ¤ Now define ϑ(x) by ∃α ∈ V(Y∗ ) p0 t ' α ∧ x = `(α) . Then CZF + ΠΣ−AC ` ∃!x[ϑ(x) ∧ ψ(x)].
u t
Theorem 8.4 Let θ1 , θ2 be Σ(gmath) sentences and let ψ(x) be a Σ(gmath) formula with at most x free. Then we have the following: (i) If CZF + REA + RDC + ΠΣW−AC ` θ1 ∨ θ2 then CZF + REA + ΠΣW−AC ` θ1 or CZF + REA + ΠΣW−AC ` θ2 . (ii) If CZF + REA + RDC + ΠΣW−AC ` ∃u ∈ ω ψ(u) then there exists a natural number n such that CZF + REA + ΠΣW−AC ` ψ(¯ n). (iii) If CZF + REA + RDC + ΠΣW−AC ` ∃xψ(x) then there is a formula ϑ(x) (with at most x free) such that CZF + REA + ΠΣW−AC ` ∃!x[ϑ(x) ∧ ψ(x)]. Proof : The proof results from that of 8.3 by replacing the reference to [21] 4.14 by reference to [21] 4.33, and using 5.21 (ii) in place of 5.21 (i). u t
References [1] P. Aczel: The Type Theoretic Interpretation of Constructive Set Theory. In: A. MacIntyre, L. Pacholski, J. Paris (eds.), Logic Colloquium ’77 North-Holland, Amsterdam (1978) 55–66. [2] P. Aczel: The type theoretic interpretation of constructive set theory: Choice principles. In: A.S. Troelstra and D. van Dalen, editors, The L.E.J. Brouwer Centenary Symposium, North Holland, Amsterdam (1982) 1–40. [3] P. Aczel: The Type Theoretic Interpretation of Constructive Set Theory: inductive definitions. In: R.B. Marcus et al. (eds), Logic, Methodology and Philosophy of Science VII, Elsevier, Amsterdam (1986) 17–49. [4] P. Aczel, M. Rathjen: Notes on Constructive Set Theory. Report No. 40 Institut Mittag-Leffler, Stockholm, Sweden (2001). [5] J. Barwise: Admissible sets and structures. Springer, Berlin (1975). [6] M. Beeson: Continuity in intuitionistic set theories. In: M Boffa, D. van Dalen, K. McAloon (eds.): Logic Colloquium ’78 (North-Holland, Amsterdam, 1979) 1–52. [7] M. Beeson: Foundations of Constructive Mathematics. Springer, Berlin (1985). [8] E. Bishop: Foundations of constructive analysis. McGraw-Hill, New York (1967). [9] R. Diaconescu: Axiom of choice and complementation. Proc. Amer. Math. Soc. 51:176–178, 1975. [10] S. Feferman: Constructive theories of functions and classes. In: In: M Boffa, D. van Dalen, K. McAloon (eds.): Logic Colloquium ’78 (North-Holland, Amsterdam, 1979) 1–52. [11] H. Friedman: Set-theoretic foundations for constructive analysis. Annals of Mathematics 105 (1977) 868-870. 29
ˇcedrov: The lack of definable witnesses and provably recursive functions in intu[12] H. Friedman, S. Sˇ itionistic set theory, Advances in Mathematics 57 (1985) 1–13. [13] P. Martin-L¨of: An intuitionistic theory of types: predicative part In: H.E. Rose and J. Sheperdson (eds.): Logic Colloquium ’73, North-Holland, Amsterdam (1975) 73–118. [14] P. Martin-L¨of: Intuitionistic Type Theory. Bibliopolis, Naples (1984). [15] Y.N. Moschovakis: Recursion in the universe of sets, mimeographed note (1976). [16] L. Moss: Power set recursion, Annals of Pure and Applied Logic 71 (1995) 247–306. [17] J. Myhill: Constructive set theory. Journal of Symbolic Logic 40 (1975) 347–382. [18] D. Normann: Set recursion. In: Generalized recursion theory II, North-Holland, Amsterdam (1978) 303–320. [19] M. Rathjen: The strength of some Martin-L¨ of type theories. Archive for Mathematical Logic (1994) 347–385. [20] M. Rathjen: Choice principles in constructive and classical set theories. To appear in: Z. Chatzidakis, P. Koepke, W. Pohlers (eds.): Proceedings of the Logic Colloquium 2002. [21] M. Rathjen: The formulae-as-classes interpretation of constructive set theory. To appear in: Proof Technology and Computation, Proceedings of the International Summer School Marktoberdorf 2003 (IOS Press, Amsterdam,2004). [22] M. Rathjen: The Disjunction and numerical existence property for constructive Zermelo-Fraenkel set theory. To appear in: L. Crosilla, P. Schuster (eds.): Proceedings of the workshop From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics, Oxford University Press. [23] G. E. Sacks: Higher Recursion Theory. Springer, Berlin (1990) [24] A. Troelstra, D. van Dalen: Constructivism in Mathematics, volume II. North Holland, Amsterdam (1988). [25] S. Tupailo: Realization of analysis into Explicit Mathematics. Journal of Symbolic Logic 66 (4) (2001) 1848–1864. [26] S. Tupailo: Realization of Constructive Set Theory into Explicit Mathematics: a lower bound for impredicative Mahlo universe. Annals of Pure and Applied Logic 120/1–3 (2003) 165–196.
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